Integral of Reciprocal is Divergent/Unbounded Above/Proof 1

Theorem

$\ds \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$

Proof

From Harmonic Series is Divergent, we have that $\ds \sum_{n \mathop = 1}^\infty \frac 1 n$ diverges to $+\infty$.

Thus from the Cauchy Integral Test:

$\ds \int_1^n \frac {\d x} x \to +\infty$

diverges.

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 13.33$